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Statistical Analysis

Statistical Analysis. Basic Calculations. Mean - Average Median - Middle Mode – Most often Range – Diff. b/w the highest and lowest. Find the mean, median, mode, and range for the following list of values: 1, 2, 4, 7 The mean is the usual average: (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5

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Statistical Analysis

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  1. Statistical Analysis

  2. Basic Calculations • Mean - Average • Median - Middle • Mode – Most often • Range – Diff. b/w the highest and lowest

  3. Find the mean, median, mode, and range for the following list of values:1, 2, 4, 7 • The mean is the usual average: (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5 • The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers. In this case, the median is the mean (the usual average) of the middle two values: (2 + 4) ÷ 2 = 6 ÷ 2 = 3 • The mode is the number that is repeated most often, but all the numbers appear only once. Then there is no mode. • The largest value is 7, the smallest is 1, and their difference is 6, so the range is 6. • mean:3.5 median: 3 mode: none range: 6

  4. A student has earned the following grades on his tests: 87, 95, 76, and 88. He wants an 85 or better overall. What is the minimum grade he must get on the last test in order to achieve that average? The unknown score is "x". Then the desired average is: • (87 + 95 + 76 + 88 + x) ÷ 5 = 85 • Multiplying through by 5 and simplifying, I get: • 87 + 95 + 76 + 88 + x = 425                       346 + x = 425 x = 79 He needs to get at least a 79 on the last test.

  5. Standard Deviation • - a measure of the range of variation (a.k.a. “the spread”) from an average group of measurements • For a normal distribution (a type of frequency distribution that is symmetrical around the mean…a.k.a. bell curve) 68% of all measurements fall within 1 sd (+/- 1sd) of the mean. • 95% of all measurements fall within 2 sd of the mean (+/- 2 sd)

  6. Standard Deviation • Using Excel • http://www.gifted.uconn.edu/siegle/research/Normal/stdexcel.htmb

  7. Standard Deviation • Ti – 83 (or more recent) edition • See handout (page 132). In-class activity • No enough calipers available. We will use rulers and tree leaves (of the same species and same plant)

  8. Hypothesis ۰To answer a statistical question, the question is translated into a hypothesis - a statement which can be subjected to test. Depending on the result of the test, the hypothesis is accepted or rejected. ۰ The hypothesis tested is known as the null hypothesis (H0). This must be a true/false statement.For every null hypothesis, there is an alternative hypothesis (HA or H1). ۰ The null hypothesis has priority and is not rejected unless there is strong evidence against it. ۰ If one of the two hypotheses is 'simpler' it is given priority so that a more 'complicated' theory is not adopted unless there is sufficient evidence against the simpler one ۰ In general, it is 'simpler' to propose that there is no difference between two sets of results than to say that there is a difference. ۰ The outcome of a hypothesis testing is "reject H0" or "do not reject H0". If we conclude "do not reject H0", this does not necessarily mean that the null hypothesis is true, only that there is insufficient evidence against H0 in favour of HA or H1. Rejecting the null hypothesis suggests that the alternative hypothesis may be true. ۰

  9. Significance Level and Critical Region • A somewhat arbitrary number usually chosen as 5%. It is actually the probability of rejecting H0 when H0 is in fact true. • We want this probability to be small but make it too small and you run the risk of NOT rejecting H0 when you should. • Critical region depends on the significance level and the type of test employed. • Found by using a statistics table • You compare your calculated statistic with the critical region. If it lies in the critical region, H0 is rejected.

  10. Hypothesis example • Examples from pages 365-366

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